Biomedical Engineering Reference
In-Depth Information
temperature. 42 In our studies of CBE in images, apparent motion
of image features has been tracked and compensated for auto-
matically as described previously, so that CBE can be measured
at each pixel in motion-compensated images. 70 Figure 13.8 shows
CBE images, that is, the energy changes relative to the reference
image at 37°C. As predicted, the energy change is both positive
(increasing) and negative (decreasing) with temperature. This
approach allows use of the whole ultrasonic image rather than
just the signals from selected scattering regions, but at the price
of possibly increasing noise in estimates of CBE.
After compensating for apparent motion in images of bovine
liver, turkey breast, and pork muscle, the mean change in back-
scattered energy at each pixel over eight image regions in all tissue
specimens was calculated with respect to a reference temperature
(37°C). As temperature increased, for some scattering regions the
CBE was positive, for others it was negative as seen in Figure 13.9
for our initial predictions, from simulations of images of scatterer
populations, and from in vitro measurements in different types
of tissue. Because the means of CBE from pixels with positive
and negative relative backscattered energy changed nearly mono-
tonically, CBE is a suitable parameter for temperature estimation.
From uniform-heating studies its accuracy and spatial resolution
appear to be suitable for temperature imaging. 73
CBE reference (37°C)
CBE @ 41°C
dB
10
10
2
2
0
0
4
4
6
−10
6
−10
5101520
5101520
CBE @ 45°C
CBE @ 50°C
10
10
2
2
0
0
4
4
6
6
−10
−10
5101520
5101520
mm
mm
FIGURE 13.8 Change in backscattered energy in ultrasound images
of bovine liver from 37 to 50°C after compensation for apparent
motion. All images were referred, pixel-by-pixel, to the energy in the
reference image at 37°C. Each color bar is in dB (similar to Figure 6,
but from a different specimen and over a larger region of interest, in
Arthur et al. 43 ).
where, as functions of temperature, α( T ) is the attenuation within
the tissue volume and η( T ) is the backscatter coefficient of the
tissue volume. Distance x is the path length in the tissue volume.
We inferred the temperature dependence of the backscatter
coefficient from the scattering cross section of a sub-wavelength
scatterer. 114,115 Neglecting the effect of the small change in the
wavenumber (<1.5%) with temperature, we assumed 91
13.4.4.1 Stochastic-Signal Framework
In our initial work, CBE from backscattered signals was com-
puted as a ratio at each pixel in the envelope-detected images of
energies at temperature T and T R . 43 It was characterized by aver-
aging ratios larger than and less than 1, denoted as positive CBE
(PCBE) and negative CBE (NCBE), which describe the increase
and decrease in the backscattered energy, respectively.
When i en is represented by a random process, computation of
the signal ratio can be modeled as a ratio between two random
variables, y T and y R : 46
2
2
2
2
ρ
cT
()
−ρ
cT
()
1
3
33
2
ρ−ρ
ρ+ρ
mms
s
s
m
+
2
ρ
cT
()
(13.8)
η
η
()
()
T
T
s
s
s
m
=
,
2
2
2
2
ρ
cT
()
()
()
−ρ
cT
1
3
33
2
ρ−ρ
ρ+ρ
R
mRms
Rs
s
m
+
y
y
z
=
T
R
(13.9)
2
ρ
cT
s
Rs
s
m
where ρ and c are the density and speed of sound of the scatterer
s and medium m . This expression applies to conditions where
the wavelength λ is larger than 2π a , where a is the radius of the
scatterer. Assuming a speed of sound of 1.5 mm/μs and a fre-
quency of 7.5 MHz, λ is 0.2 mm, which means that Equation 13.7
using Equation 13.8 applies to scatterers smaller than 30 μm.
Changes in backscattered energy were modeled assuming that
the scattering potential of the volume was proportional to the
scattering cross section of sub-wavelength scatterers. We pre-
dicted with this model that the change in backscattered energy
could increase or decrease depending on what type of inhomo-
geneity caused the scattering. These calculations suggested that
the change in backscattered energy could vary depending on the
type of scatterers in a given tissue region.
In 1D studies, we showed that it is possible to isolate and mea-
sure backscattered energy from individual scattering regions,
and that measured CBE was nearly monotonically dependent on
where y R and y T are the random variables representing the
B-scans at the reference and current temperatures. The ratio,
z , is also a random variable whose probability density function
(PDF), f Z ( z ), is determined by the joint distribution of ( y R , y T ): 116
fz
()
=
||
yf
(,
yyzdy
)
,
(13.10)
Z
R
YY
RR
R
RT
−∞
where y R , y T , and z > 0.
The computation of PCBE in our initial work can be written
as
1
z
N
k
1
kkz
N
N
∈>
{|
1}
PCBE
=
z
=
,
k
k
N
+
+
kkz
∈>
{|
1}
k
where N is the number of pixels in image and N + is the number of
ratio pixels with value larger than 1. Assuming z k 's are independent,
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